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Infinite Series and Partial Sums

Calculus 2 · Chapter 6: Sequences & Introduction to SeriesSection 6.2 · lesson 28 of 549 min · free

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In this lesson

What it means to add infinitely many numbers.

  • Define a series' sum as the limit of its partial sums
  • Distinguish convergence from the two kinds of divergence
  • See why the harmonic series diverges anyway
  • Apply the Divergence Test - and know what it cannot do

Transcript

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MAT-265 - Calculus 2 · Section 6.2

Infinite Series and Partial Sums

Adding infinitely many numbers is not an operation that arithmetic defines. You can add two numbers. You can add a thousand. But you cannot literally add infinitely many, one at a time, and finish. So we have to define what it means. And the definition is beautifully simple. Add finitely many at a time, and watch where the running total goes. The nth partial sum is just the first n terms added up. s one is a one. s two is a one plus a two. And in general s n is the sum from k equals one to n, of a k. Each s n is an ordinary finite sum. Nothing mysterious. But as n increases, these partial sums form their own sequence. And the behavior of that sequence — a sequence, which we just learned to analyze in section six point one — defines the sum of the series.

The Definition

A Series Is a Sequence in Disguise

Here's the formal definition. The sum of the infinite series, sum from n equals one to infinity of a n, is defined as the limit of its sequence of partial sums. If that limit exists and equals a finite number L, the series converges to L. If the limit does not exist — including the case where the partial sums grow without bound — the series diverges. This definition is the cornerstone of the entire series unit. Every later test — comparison, ratio, integral, and all the rest — is ultimately a tool for determining the behavior of this limit, without computing it directly. So understanding that the sum of a series means the limit of its partial sums is essential before any of those tests will make sense.

Worked Example 1

Zeno's Paradox, Resolved

Let's do the most famous example there is. Show that the sum of one over two to the n converges, and find its sum. That's one half, plus one quarter, plus one eighth, and so on forever. Compute several partial sums to see the pattern. s one is one half. s two is one half plus one quarter, which is three quarters. s three is three quarters plus one eighth. Seven eighths. s four is fifteen sixteenths. See the pattern? Each partial sum is two to the n, minus one, over two to the n. Which is one, minus one over two to the n. Now take the limit as n goes to infinity. One over two to the n goes to zero. So the limit is one minus zero. One. The limit exists and equals one. So the series converges, and its sum is exactly one. And this is the rigorous resolution of Zeno's paradox. The partial sums approach one, but never exceed it. Infinitely many steps. A finite total distance.

Figure 6.2.1

Two Kinds of Behaviour

Look at the two pictures, because they show the two behaviors clearly. On the left, the partial sums of the sum of one over two to the n. They climb toward the dashed line at L equals one, and settle there. The series converges. On the right, the partial sums of the harmonic series — the sum of one over n. They keep rising. And rising. They never level off. So the series diverges — even though its individual terms shrink toward zero. Hold that thought, because it's the most important lesson in this section. Now, divergence takes more than one form. The partial sums may grow without bound, to plus or minus infinity. Or they may oscillate, without ever settling. Consider the sum of minus one to the n plus one. That's one, minus one, plus one, minus one, forever. List the partial sums. s one is one. s two is zero. s three is one. s four is zero. They alternate one, zero, one, zero, forever. Never approaching a single value. So the limit of s n does not exist, and the series diverges. Note that it doesn't run off to infinity. It simply fails to settle. Which is enough for divergence.

The Harmonic Series

Shrinking Terms Are Not Enough

The harmonic series deserves its own moment, because it's the standard warning of this entire chapter. Its terms are one, one half, one third, one quarter, and so on. They shrink steadily toward zero. Which makes it extremely tempting to expect convergence. Yet its partial sums grow without bound. It diverges. So here's the question worth sitting with. The sum of one over two to the n converges. The harmonic series diverges. Yet both have terms that shrink toward zero. What distinguishes them? The terms of one over two to the n shrink fast. Each is exactly half the previous one. So they add up to a finite total. The harmonic terms shrink too slowly. Although each individual term is small, there are enough of them at each scale to keep the running total climbing forever. So the lesson is this. Convergence depends not just on whether terms approach zero — but on how quickly. Terms shrinking to zero is necessary. It is not sufficient.

The Divergence Test

A One-Way Tool

The harmonic series shows that shrinking terms do not guarantee convergence. But the reverse implication does hold. And it gives a quick, powerful test for divergence. If the terms do not shrink to zero, the series cannot possibly converge. That's the Divergence Test, sometimes called the nth-Term Test. If the limit of a n is not zero — or if that limit does not exist — then the series diverges. The reasoning is short, and worth understanding, because it follows directly from the partial-sum definition. Suppose the series converges to a sum L. So s n approaches L. Then the partial sum one step earlier also approaches the same limit — shifting the index by one doesn't change a sequence's limit. Now observe that the nth term is the difference of consecutive partial sums. a n equals s n minus s n minus one. Take the limit of both sides. You get L minus L. Zero. So if a series converges, its terms must approach zero. Take the contrapositive, and you have the test exactly. If the terms don't approach zero, the series cannot converge. Let's use it. Apply the test to the sum of n over two n plus one. The limit of the terms is one half. Which is not zero. So the test applies, and the series diverges. Intuitively, the series is trying to add infinitely many numbers each close to one half. Its partial sums must grow without bound. Now a second one. Apply the test to the sum of one over n squared. The limit of the terms is zero. So the test draws no conclusion. It's inconclusive. In fact this series does converge — later tests will confirm that. But the Divergence Test alone cannot establish it.

Key Takeaways

What to Carry Forward

That brings us to the critical warning about this test, and it's where students lose the most marks. The Divergence Test is a one-way tool. It can only confirm divergence. It can never confirm convergence. If the limit of a n is zero, the test is inconclusive. The series might converge, or might diverge, and some other method is needed. The harmonic series is the perfect illustration. Its terms satisfy the limit of one over n equals zero. Yet it diverges. A common error is to see terms going to zero and wrongly declare convergence. The test simply says nothing in that case. A zero term-limit sends the problem to other tests. Never to a verdict of convergence. Five takeaways. An infinite series is defined through its partial sums, and the sum of the series is the limit of those partial sums, when that limit exists. The series converges if the partial sums approach a finite limit, and diverges otherwise — whether they grow without bound, or merely oscillate. The harmonic series diverges despite its terms shrinking to zero. Terms approaching zero is necessary, but not sufficient. The Divergence Test: if the limit of a n is not zero, or fails to exist, the series diverges. And it follows from a n equals s n minus s n minus one. And the test only proves divergence. If the limit of a n is zero, it's inconclusive, and a different test is required. Next up is section six point three — Geometric Series. The one family whose sum we can write down exactly.

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FAQ

Questions students ask us

About Infinite Series and Partial Sums

What does it mean for an infinite series to converge?
An infinite series converges when its sequence of partial sums approaches a finite limit, and that limit is defined as the sum of the series. The nth partial sum is s_n = \sum_{k=1}^{n} a_k, an ordinary finite sum. If \lim_{n\to\infty} s_n = L exists and is finite, then \sum_{n=1}^{\infty} a_n = L. Otherwise the series diverges, either because the partial sums grow without bound or because they oscillate without settling.
How do you find the sum of 1/2 + 1/4 + 1/8 + ... using partial sums?
The sum is exactly 1, found by taking the limit of the partial sums. Computing a few gives s_1 = \frac{1}{2}, s_2 = \frac{3}{4}, s_3 = \frac{7}{8}, and s_4 = \frac{15}{16}. The pattern is s_n = \frac{2^n - 1}{2^n} = 1 - \frac{1}{2^n}. Since \frac{1}{2^n} \to 0, the limit is 1. This resolves Zeno's paradox: infinitely many steps can cover a finite total distance.
Why does the harmonic series diverge if its terms go to zero?
The harmonic series \sum \frac{1}{n} diverges because its terms shrink too slowly for the partial sums to level off. The partial sums keep rising without bound. By contrast, the terms of \sum \frac{1}{2^n} halve at each step, which is fast enough to give a finite total. Convergence depends on how quickly the terms approach zero. Terms approaching zero is necessary for convergence, but it is not sufficient.
Does the Divergence Test prove a series converges if the terms go to zero?
No, the Divergence Test can only prove divergence. It is inconclusive when \lim_{n\to\infty} a_n = 0. If the limit is nonzero or does not exist, the series diverges. For example, \sum \frac{n}{2n+1} diverges because its terms approach \frac{1}{2}. However, \sum \frac{1}{n^2} (which converges) and \sum \frac{1}{n} (which diverges) both have terms tending to zero, so another test is needed to decide.

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